Sure, let's analyze the problem step-by-step:
Given:
1. [tex]\( f(x) = \sqrt{mx} \)[/tex]
2. [tex]\( g(x) = m\sqrt{x} \)[/tex]
We need to identify the conditions under which the ranges (outputs) of both functions are the same and interpret what this implies about the value of [tex]\( m \)[/tex].
### Step 1: Analyze the range of [tex]\( f(x) = \sqrt{mx} \)[/tex]
- For [tex]\( f(x) \)[/tex] to be defined, the expression inside the square root must be non-negative: [tex]\( mx \ge 0 \)[/tex].
- Since [tex]\( x \ge 0 \)[/tex] for the square root function to be defined in real numbers:
- If [tex]\( m > 0 \)[/tex], [tex]\( f(x) = \sqrt{mx} \)[/tex] will yield non-negative values, i.e., any non-negative real number as [tex]\( x \)[/tex] ranges over all non-negative values.
### Step 2: Analyze the range of [tex]\( g(x) = m\sqrt{x} \)[/tex]
- For [tex]\( g(x) \)[/tex] to be defined, the expression inside the square root (i.e., [tex]\( x \)[/tex]) must also be non-negative: [tex]\( x \ge 0 \)[/tex].
- If [tex]\( m > 0 \)[/tex], [tex]\( g(x) = m\sqrt{x} \)[/tex] will yield non-negative values, specifically any non-negative real number as [tex]\( x \)[/tex] varies over all non-negative values.
- If [tex]\( m < 0 \)[/tex], [tex]\( g(x) = m\sqrt{x} \)[/tex] will yield non-positive values, specifically any non-positive real number because [tex]\( \sqrt{x} \ge 0 \)[/tex] for [tex]\( x \ge 0 \)[/tex].
### Step 3: Compare the ranges of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]
- For the ranges of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] to be the same:
- If [tex]\( m > 0 \)[/tex], [tex]\( f(x) = \sqrt{mx} \)[/tex] and [tex]\( g(x) = m\sqrt{x} \)[/tex] both yield any non-negative real number.
- If [tex]\( m < 0 \)[/tex], [tex]\( f(x) = \sqrt{mx} \)[/tex] yields any non-negative real number, but [tex]\( g(x) = m\sqrt{x} \)[/tex] yields any non-positive real number.
### Conclusion:
The ranges of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] can only be the same when [tex]\( m > 0 \)[/tex].
Thus, the statement that is true about the value of [tex]\( m \)[/tex] is:
[tex]\[ m \text{ can be any positive real number.} \][/tex]
